MODERN COSMOLOGY

Dynamics of structure formation 57

These expressions assume pure dark matter, which is unrealistic. At least
for CDM models, a non-zero baryonic density lowers the apparent dark-matter
density parameter. We can define an apparent shape parameterfor the transfer
function:
q≡(k/hMpc−^1 )/,

and=
hin a model with zero baryon content. This parameter was originally
defined by Efstathiouet al(1992), in terms of a CDM model with
B= 0 .03.
Peacock and Dodds (1994) showed that the effect of increasing
Bwas to
preserve the CDM-style spectrum shape, but to shift to lower values of.This
shift was generalized to models with
=1 by Sugiyama (1995):

=
hexp[−
B( 1 +

√

2 h/
)].

Note the oscillations inT(k)for high baryon content; these can be significant even
in CDM-dominated models when working with high-precision data. Eisenstein
and Hu (1998) are to be congratulated for their impressive persistence in finding
an accurate fitting formula that describes these wiggles. This is invaluable for
carrying out a search of a large parameter space. An interesting question is
whether these ‘wiggles’ survive evolution into the nonlinear regime: Meiksinet al
(1999) showed that most do not, but that observable signatures of baryons remain
on large scales.

2.6.5 The spherical model

An overdense sphere is a very useful nonlinear model, as it behaves in exactly
the same way as a closed sub-universe. The density perturbation needs not be a
uniform sphere: any spherically symmetric perturbation will clearly evolve at a
given radius in the same way as a uniform sphere containing the same amount of
mass. In what follows, therefore, density refers to themeandensity inside a given
sphere. The equations of motion are the same as for the scale factor, and we can
therefore write down the cycloid solution immediately. For a matter-dominated
universe, the relation between the proper radius of the sphere and time is

r=A( 1 −cosθ)

t=B(θ−sinθ),

andA^3 =GMB^2 , just fromr ̈=−GM/r^2. Expanding these relations up to
orderθ^5 givesr(t)for smallt:

r

A

2

(

6 t

B

) 2 / 3 [

1 −

1

20

(

6 t

B

) 2 / 3 ]

,

and we can identify the density perturbation within the sphere:

δ

3

20

(

6 t

B

) 2 / 3

.